Mean field games (MFGs) and mean field control (MFC) provide a mathemat-
ical framework for modeling large controlled interacting particle systems, with
applications ranging from financial mathematics to biological neural networks. In
the first part of this talk, I will give an accessible introduction to MFGs and MFC.
I will then show how the infinite-dimensional case arises naturally in applications,
highlight its specific analytical challenges, and provide an overview of my recent
results.
In the second part of the talk, I will detail how we address these challenges in
recent joint work with A. Święch, where we investigate an MFG posed in a Hilbert
space and driven by degenerate noise. We study the associated MFG system
consisting of a Hamilton–Jacobi–Bellman (HJB) equation for the value function
coupled with a nonlinear Fokker–Planck (FP) equation for the distribution of the
particles, both governed by a degenerate Kolmogorov operator. This degeneracy
introduces significant analytical challenges. In particular, the HJB equation is
treated in the viscosity sense, while the FP equation is interpreted in a suitable
weak formulation. Proving the uniqueness of solutions to the FP equation becomes
particularly delicate. Under appropriate structural assumptions, we establish the
well-posedness of the MFG system.

Ballistic deposition is a model of interface growth introduced by Vold in 1959,
which has remained largely mathematically intractable. It is defined as a process
of vertically falling blocks that stick to the top, the right or the left of growing
columns. We introduce a variant, the one-sided ballistic deposition model, in
which vertically falling blocks can only stick to the top or the upper right corner
of growing columns. We establish that in dimension d=2, strong KPZ universality
holds near the time axis, proving that the fluctuations of the height function there
are given by the Tracy-Widom GUE distribution. The proof is based on a graphical
construction of the process in terms of a last passage percolation model. This talk
is based on a joint work with Pablo Groisman, Santiago Saglietti and Sebastián
Zaninovich.

Mean field type control problems lead to Hamilton–Jacobi–Bellman (HJB) equations on the space of probability measures. In this talk, we will present a probabilistic approach to constructing classical solutions through forward-backward stochastic differential equations (FBSDEs), allowing for degenerate and unbounded diffusion coefficients. Starting with linear dynamics, we will explain how convexity ensures the global well-posedness of the associated FBSDE system, and how variational equations establish the regularity of the value function needed for the HJB equation. We will then extend the framework to jump-diffusion models, where the associated HJB equation is an integro-partial differential equation. Finally, we will consider the case where a backward equation forms part of the state.